Ind-completion

In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category C. The objects in this ind-completed category, denoted Ind(C), are known as direct systems, they are functors from a small filtered category I to C.

The dual concept is the pro-completion, Pro(C).

Definitions

Filtered categories

Direct systems depend on the notion of filtered categories. For example, the category N, whose objects are natural numbers, and with exactly one morphism from n to m whenever ${\displaystyle n\leq m}$, is a filtered category.

Direct systems

A direct system or an ind-object in a category C is defined to be a functor

${\displaystyle F:I\to C}$

from a small filtered category I to C. For example, if I is the category N mentioned above, this datum is equivalent to a sequence

${\displaystyle X_{0}\to X_{1}\to \cdots }$

of objects in C together with morphisms as displayed.

The ind-completion

Ind-objects in C form a category ind-C, and pro-objects form a category pro-C. The definition of pro-C is due to Grothendieck (1960).[1]

Two ind-objects

${\displaystyle F:I\to C}$

and

${\textstyle G:J\to C}$ determine a functor

Iop x J ${\displaystyle \to }$ Sets,

namely the functor

${\displaystyle \operatorname {Hom} _{C}(F(i),G(j)).}$

The set of morphisms between F and G in Ind(C) is defined to be the colimit of this functor in the second variable, followed by the limit in the first variable:

${\displaystyle \operatorname {Hom} _{\operatorname {Ind} {\text{-}}C}(F,G)=\lim _{i}\operatorname {colim} _{j}\operatorname {Hom} _{C}(F(i),G(j)).}$

More colloquially, this means that a morphism consists of a collection of maps ${\displaystyle F(i)\to G(j_{i})}$ for each i, where ${\displaystyle j_{i}}$ is (depending on i) large enough.

Relation between C and Ind(C)

The final category I = {*} consisting of a single object * and only its identity morphism is an example of a filtered category. In particular, any object X in C gives rise to a functor

${\displaystyle \{*\}\to C,*\mapsto X}$

and therefore to a functor

${\displaystyle C\to \operatorname {Ind} (C),X\mapsto (*\mapsto X).}$

This functor is, as a direct consequence of the definitions, fully faithful. Therefore Ind(C) can be regarded as a larger category than C.

Conversely, there need in general not be a natural functor

${\displaystyle \operatorname {Ind} (C)\to C.}$

However, if C possesses all filtered colimits (also known as direct limits), then sending an ind-object ${\displaystyle F:I\to C}$ (for some filtered category I) to its colimit

${\displaystyle \operatorname {colim} _{I}F(i)}$

does give such a functor, which however is not in general an equivalence. Thus, even if C already has all filtered colimits, Ind(C) is a strictly larger category than C.

Objects in Ind(C) can be thought of as formal direct limits, so that some authors also denote such objects by

${\displaystyle {\text{“}}\varinjlim _{i\in I}{\text{'' }}F(i).}$

Universal property of the ind-completion

The passage from a category C to Ind(C) amounts to freely adding filtered colimits to the category. This is why the construction is also referred to as the ind-completion of C. This is made precise by the following assertion: any functor ${\displaystyle F:C\to D}$ taking values in a category D which has all filtered colimits extends to a functor ${\displaystyle Ind(C)\to D}$ which is uniquely determined by the requirements that its value on C is the original functor F and such that it preserves all filtered colimits.

Basic properties of ind-categories

Compact objects

Essentially by design of the morphisms in Ind(C), any object X of C is compact when regarded as an object of Ind(C), i.e., the corepresentable functor

${\displaystyle \operatorname {Hom} _{\operatorname {Ind} (C)}(X,-)}$

preserves filtered colimits. This holds true no matter what C or the object X is, in contrast to the fact that X need not be compact in C. Conversely, any compact object in Ind(C) arises as the image of an object in X.

A category C is called compactly generated, if it is equivalent to ${\displaystyle \operatorname {Ind} (C_{0})}$ for some small category ${\displaystyle C_{0}}$. The ind-completion of the category FinSet of finite sets is the category of all sets. Similarly, if C is the category of finitely generated groups, ind-C is equivalent to the category of all groups.

Recognizing ind-completions

These identifications rely on the following facts: as was mentioned above, any functor ${\displaystyle F:C\to D}$ taking values in a category D that has all filtered colimits, has an extension

${\displaystyle {\tilde {F}}:\operatorname {Ind} (C)\to D,}$

which is unique up to equivalence. First, this functor ${\displaystyle {\tilde {F}}}$ is essentially surjective if any object in D can be expressed as a filtered colimits of objects of the form ${\displaystyle F(c)}$ for appropriate objects c in C. Second, ${\displaystyle {\tilde {F}}}$ is fully faithful if and only if the original functor F is fully faithful and if F sends arbitrary objects in C to compact objects in D.

Applying these facts to, say, the inclusion functor

${\displaystyle F:\operatorname {FinSet} \subset \operatorname {Set} ,}$

the equivalence

${\displaystyle \operatorname {Ind} (\operatorname {FinSet} )\cong \operatorname {Set} }$

expresses the fact that any set is the filtered colimit of finite sets (for example, any set is the union of its finite subsets, which is a filtered system) and moreover, that any finite set is compact when regarded as an object of Set.

The pro-completion

Like other categorical notions and constructions, the ind-completion admits a dual known as the pro-completion: the category Pro(C) is defined in terms of ind-object as

${\displaystyle \operatorname {Pro} (C):=\operatorname {Ind} (C^{op})^{op}.}$

Therefore, the objects of Pro(C) are inverse systems or pro-objects in C. By definition, these are direct system in the opposite category ${\displaystyle C^{op}}$ or, equivalently, functors

${\displaystyle F:I\to C}$

from a cofiltered category I.

Examples of pro-categories

While Pro(C) exists for any category C, several special cases are noteworthy because of connections to other mathematical notions.

• If C is the category of finite groups, then pro-C is equivalent to the category of profinite groups and continuous homomorphisms between them.
• The process of endowing a preordered set with its Alexandrov topology yields an equivalence of the pro-category of finite preordered sets, ${\displaystyle \operatorname {Pro} (\operatorname {PoSet} ^{\text{fin}})}$, with the category of spectral topological spaces and quasi-compact morphisms.
• Stone duality asserts that the pro-category ${\displaystyle \operatorname {Pro} (\operatorname {FinSet} )}$ of the category of finite sets is equivalent to the category of Stone spaces.[2]

The appearance of topological notions in these pro-categories can be traced to the equivalence, which is itself a special case of Stone duality,

${\displaystyle \operatorname {FinSet} ^{op}=\operatorname {FinBool} }$

which sends a finite set to the power set (regarded as a finite Boolean algebra). The duality between pro- and ind-objects and known description of ind-completions also give rise to descriptions of certain opposite categories. For example, such considerations can be used to show that the opposite category of the category of vector spaces (over a fixed field) is equivalent to the category of linearly compact vector spaces and continuous linear maps between them.[3]

Applications

Pro-completions are less prominent than ind-completions, but applications include shape theory. Pro-objects also arise via their connection to pro-representable functors, for example in Grothendieck's Galois theory, and also in Schlessinger's criterion in deformation theory.

Tate object are a mixture of ind- and pro-objects.

Infinity-categorical variants

The ind-completion (and, dually, the pro-completion) has been extended to ∞-categories by Lurie (2009).

Notes

1. C.E. Aull; R. Lowen (31 December 2001). Handbook of the History of General Topology. Springer Science & Business Media. p. 1147. ISBN 978-0-7923-6970-7.
2. Johnstone (1982, §VI.2)
3. Bergman & Hausknecht (1996, Prop. 24.8)

References

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